An Ultrametric State Space with a Dense Discrete Overlap Distribution: Paperfolding Sequences

Paperfolding Sequences, Paperfolding Curves and Local Isomorphism

For each integer n, an n-folding curve is obtained by folding n times a strip of paper in two, possibly up or down, and unfolding it with right angles. Generalizing the usual notion of infinite folding curve, we define complete folding curves as the curves without endpoint which are unions of increasing sequences of n-folding curves for n integer. We prove that there exists a standard way to ex...

Percolation in an ultrametric space

We study percolation in the hierarchical lattice of order N where the probability of connection between two points separated by distance k is of the form ck/N , δ > −1. Since the distance is an ultrametric, there are significant differences with percolation in the Euclidean lattice. We consider three regimes: δ < 1, where percolation occurs, δ > 1, where it does not occur, and δ = 1 which is th...

Optimal Description of Automatic Paperfolding Sequences

The class of 2-automatic paperfolding sequences corresponds to the class of ultimately periodic sequences of unfolding instructions. We rst show that a paper-folding sequence is automatic ii it is 2-automatic. Then we provide families of minimal nite-state automata, minimal uniform tag sequences and minimal substitutions describing automatic paperfolding sequences, as well as a family of algebr...

Diffusion in an ultrametric space : a simple case

Admissibility analysis for discrete-time singular systems with time-varying delays by adopting the state-space Takagi-Sugeno fuzzy model

This paper is pertained with the problem of admissibility analysis of uncertain discrete-time nonlinear singular systems by adopting the state-space Takagi-Sugeno fuzzy model with time-delays and norm-bounded parameter uncertainties. Lyapunov Krasovskii functionals are constructed to obtain delay-dependent stability condition in terms of linear matrix inequalities, which is dependent on the low...